Can Neural Networks self-optimize?
Since a neural network does iteratively learn its own weights I assume you mean the structure of the neural network - the number of layers and nodes per layer.

If what I said above was your question, then yes, it most definitely is being explored.
Even when a neural network is allowed to learn its own structure it still needs to be suited to a specific problem. 
Google wrote a blog where they used machine learning to explore the architecture of a neural network calling it the AutoML project where the blog claimed "To make this process of designing machine learning models much more accessible, we’ve been exploring ways to automate the design of machine learning models."
It was found that AutoML can design small neural networks that perform on par with neural networks designed by human experts, but these results were constrained to small academic datasets like CIFAR-10, and Penn Treebank.

This study was further extended to more challenging problems and datasets in another blog where they proposed NASNet 

Both these blogs were written in 2017 and I'm not very sure how much progress has been made on this since, but I hope these blogs helped!